# Full text of "Treatise On Analysis Vol-Ii"

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```4    HAAR MEASURE ON A QUOTIENT GROUP       267

(14.4.3)    (i)   /// is any mapping of G into R, the number f * f(x£) rfa'(£>

depends only on x = n(x). Iffe ./(G), the function xh-»J*/(*0 doc'(£) belongs
to J*XG"), and we have

(14.4.3.1)              J/(x) da(x) = pa"(x) J */(x

(ii)    7/*/w any mapping ofG into R, then

(14.4.3.2)                f */(x) da(x) ^ f da"(x) [ */(x{) da'({),

J                      */            J

(14.4.3.3)                f /(x) da(x) £ f da"(Jc) f/(x£) da'«).
J*                    J*         J*

(iii)   Let N be an ^-negligible set in G, and let M ie the set of all xeG such
that x~"l(N n xG') is not a' -negligible in G'. Then n(M) is a" '-negligible.

The proofs follow step by step those of (13.21.3), (13.21.4), and (13.21.5),
using (14.4.1) and the left invariance of a'.

(14.4.4)    Let u be an ^-measurable mapping of G into a topological space E,
and let N be the set of all xeG such that the partial mapping % H-» w(x£) is not
a! -measurable. Then 7r(N) is a" -negligible.

The proof follows the lines of the proof of (1 3.21 .6) using (1 4.4.3) in place
of (13.21.5).

(14.4.5)    Let f be an a-integrable mapping ofG into R, and let N be the set of
all xeG such that the partial mapping £\-+f(x§ ofG' into R is not a'-integ-

rable. Then rc(N) is VL" -negligible \ for each x <£ 7r(N), the number I f(x£) da'(£)
is the same for all points x e x; the function XH~> j/(x<i;) da'(0» defined almost
everywhere on G", is a"-integrable; and the formula (14.4.2) is valid.

Here again, we have only to follow step by step the proof of the theorem of
Lebesgue-Fubini (13.21.7), using (14.4.3) and (14.4.4).

(14.4.6)    Let u be an automorphism of the topological group G such that
u(Gf) = G'. Let u' be the restriction ofu to G', and let u" be the automorphism
ofG" induced by u (so that u"(n(x)) = n(u(x)) for all x e G). Then modG(w) =
modG<(t/') • modG»(w"). balls with center 0 and contained in A,
```